There are two possible ways to get rid of an instagram account: by disabling it for a while and by deleting it permanently. If you temporarily disable your account, your profile, photos, comments and likes will be hidden until you are ready to reactivate it. But If you delete your account, your profile, photos, videos, comments, likes and followers will be permanently removed. If you would just like to take a break, you can temporarily disable your account instead.

After you delete your account, you can not sign up again with the same username and you can't add that username to another account. Note that Instagram doesn't reactivate deleted accounts.

To temporarily disable your account:

Go to instagram.com from a mobile browser or computer and login to your account. You can not temporarily disable your account within the Instagram app.

Click on in the top right and then select Edit Profile.

Scroll through, then click on 'Temporarily disable my account' in the bottom right.

Pick an option from the drop-down menu beside 'Why are you disabling your account?' and re-enter your password. The option to disable your account only appears after you have selected a reason from the menu.

Click on Temporarily Disable Account.

You will have to log into your account to temporarily disable it.
If you do not want to disable your account but want to change who can see it, you can set your posts to private or you can block people.

To permanently delete your account:

Go to the' Delete Your Account' page. If you are not already logged into Instagram on the web, you'll be asked to log in first. You can not delete your account from the Instagram app.

Pick an option from the drop-down menu next to 'Why are you deleting your account?' and re-enter your password. The option to permanently delete your account will appear only after you have selected a reason from the menu.

Click on Permanently delete my account.

If you would like to delete a different account, click the username in the top-right of the Delete Your Account page, click on next to username and click on Log Out. Log back in with the account you want to delete and follow the instructions above.

For security reasons, Instagram can not delete an account for you. You will need to be able to login to your account to delete it.